Fit a p*/ERG Model Using a Logistic Approximation
Fits a p*/ERG model to the graph in dat containing the effects listed in effects. The result is returned as a glm object.
pstar(dat, effects=c("choice", "mutuality", "density", "reciprocity",
"stransitivity", "wtransitivity", "stranstri", "wtranstri",
"outdegree", "indegree", "betweenness", "closeness",
"degcentralization", "betcentralization", "clocentralization",
"connectedness", "hierarchy", "lubness", "efficiency"),
attr=NULL, memb=NULL, diag=FALSE, mode="digraph")dat |
a single graph |
effects |
a vector of strings indicating which effects should be fit. |
attr |
a matrix whose columns contain individual attributes (one row per vertex) whose differences should be used as supplemental predictors. |
memb |
a matrix whose columns contain group memberships whose categorical similarities (same group/not same group) should be used as supplemental predictors. |
diag |
a boolean indicating whether or not diagonal entries (loops) should be counted as meaningful data. |
mode |
|
p* (also called the Exponential Random Graph (ERG) family) is an exponential family specification for network data. Under p*, it is assumed that
p(G=g) propto exp(beta_0 gamma_0(g) + beta_1 gamma_1(g) + …)
for all g, where the betas represent real coefficients and the gammas represent functions of g. Unfortunately, the unknown normalizing factor in the above expression makes evaluation difficult in the general case. One solution to this problem is to operate instead on the edgewise log odds; in this case, the p* MLE can be approximated by a logistic regression of each edge on the differences in the gamma scores induced by the presence and absence of said edge in the graph (conditional on all other edges). It is this approximation (known as autologistic regression, or maximum pseudo-likelihood estimation) which is employed here.
Using the effects argument, a range of different potential parameters can be estimated. The network measure associated with each is, in turn, the edge-perturbed difference in:
choice: the number of edges in the graph (acts as a constant)
mutuality: the number of reciprocated dyads in the graph
density: the density of the graph
reciprocity: the edgewise reciprocity of the graph
stransitivity: the strong transitivity of the graph
wtransitivity: the weak transitivity of the graph
stranstri: the number of strongly transitive triads in the graph
wtranstri: the number of weakly transitive triads in the graph
outdegree: the outdegree of each actor (|V| parameters)
indegree: the indegree of each actor (|V| parameters)
betweenness: the betweenness of each actor (|V| parameters)
closeness: the closeness of each actor (|V| parameters)
degcentralization: the Freeman degree centralization of the graph
betcentralization: the betweenness centralization of the graph
clocentralization: the closeness centralization of the graph
connectedness: the Krackhardt connectedness of the graph
hierarchy: the Krackhardt hierarchy of the graph
efficiency: the Krackhardt efficiency of the graph
lubness: the Krackhardt LUBness of the graph
(Note that some of these do differ somewhat from the common p* parameter formulation, e.g. quantities such as density and reciprocity are computed as per the gden and grecip functions rather than via the unnormalized "choice" and "mutual" quantities one often finds in the p* literature.) Please do not attempt to use all effects simultaneously!!! In addition to the above, the user may specify a matrix of individual attributes whose absolute dyadic differences are to be used as predictors, as well as a matrix of individual memberships whose dyadic categorical similarities (same/different) are used in the same manner.
Although the p* framework is quite versatile in its ability to accommodate a range of structural predictors, it should be noted that the substantial collinearity of many of the standard p* predictors can lead to very unstable model fits. Measurement and specification errors compound this problem; thus, it is somewhat risky to use p* in an exploratory capacity (i.e., when there is little prior knowledge to constrain choice of parameters). While raw instability due to multicollinearity should decline with graph size, improper specification will still result in biased coefficient estimates so long as an omitted predictor correlates with an included predictor. Caution is advised.
A glm object
Estimation of p* models by maximum pseudo-likelihood is now known to be a dangerous practice. Use at your own risk.
Carter T. Butts buttsc@uci.edu
Anderson, C.; Wasserman, S.; and Crouch, B. (1999). “A p* Primer: Logit Models for Social Networks. Social Networks, 21,37-66.
Holland, P.W., and Leinhardt, S. (1981). “An Exponential Family of Probability Distributions for Directed Graphs.” Journal of the American statistical Association, 81, 51-67.
Wasserman, S., and Pattison, P. (1996). “Logit Models and Logistic Regressions for Social Networks: I. An introduction to Markov Graphs and p*.” Psychometrika, 60, 401-426.
#Create a graph with expansiveness and popularity effects
in.str<-rnorm(20,0,3)
out.str<-rnorm(20,0,3)
tie.str<-outer(out.str,in.str,"+")
tie.p<-apply(tie.str,c(1,2),function(a){1/(1+exp(-a))})
g<-rgraph(20,tprob=tie.p)
#Fit a model with expansiveness only
p1<-pstar(g,effects="outdegree")
#Fit a model with expansiveness and popularity
p2<-pstar(g,effects=c("outdegree","indegree"))
#Fit a model with expansiveness, popularity, and mutuality
p3<-pstar(g,effects=c("outdegree","indegree","mutuality"))
#Compare the model AICs -- use ONLY as heuristics!!!
extractAIC(p1)
extractAIC(p2)
extractAIC(p3)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.